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In this paper, we give a lower bound for the maximum size of a   k  ‐colored sum‐free set in    Z  m  n   , where    k  ⩾  3    and    m  ⩾  2    are fixed and   n   tends to infinity. If   m   is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a   k  ‐colored sum‐free set in    Z  m  n   . This generalizes a result of Kleinberg–Sawin–Speyer for the case    k  =  3    and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg–Sawin–Speyer. Both of these generalizations require several key new ideas.

A lower bound for the k ‐multicolored sum‐free problem in Z m n